Navigation of Miniature Legged Robots Using a New Template

Konstantinos Karydis, Yan Liu, Ioannis Poulakakis, and Herbert G. Tanner

Abstract— This paper contributes to the area of miniature legged robots by investigating how a recently introduced bio- inspired template for such robots can be used for navigation. The model is simple and intuitive, and capable of capturing the salient features of the horizontal-plane behavior of an eight- legged miniature robot. We validate that the model can be combined with readily available navigation techniques, and then use it to plan the motion of the eight-legged miniature robot, which is tasked to crawl at low speeds, in obstacle-cluttered environments.

I.INTRODUCTION Legged robots have the potential to traverse a wide range Fig. 1. The OctoRoACH, designed at the University of California, Berkeley. of challenging terrains, where their wheeled counterparts It is 130 mm-long, weights 35 g, and reaches a maximum speed of 0.5 m/s. may not be successful. Miniature legged robots, in particular, can also reach areas where larger ones cannot fit. In addition, they can be manufactured in a fast and relatively inexpensive Sliding Spring Leg (SSL) model [18] includes the sliding manner, thus allowing for deployment in large numbers. effects of the leg-ground interaction in hexapedal robots. These features create new opportunities in applications in- However, the dynamic nature of these models presents volving building and pipe inspection, search-and-rescue, as challenges to navigation at low crawling speeds. In this well as Intelligence, Surveillance, and Reconnaissance (ISR). regime, surface forces dominate over inertia effects [19], These new opportunities have promoted the development [20]—yet, detailed ground interaction descriptions for inte- of a variety of bio-inspired multi-legged robots of small gration into a dynamical model are unavailable [21], and scale. Examples include the cockroach-inspired hexapod [1], the connection between model parameters and robot control the three-spoke rimless wheeled Mini-Whegs [2], [3], the parameters is unclear [8]. To tackle these issues, a kinematic 3D-printed robots STAR [4] and PSR [5], and i-Sprawl [6]. template called the Switching Four-bar Mechanism (SFM) is Using the Smart Composite Microstructure (SCM) tech- introduced in [22] and studied in [23]. Motivated by the foot- nique [7], several minimally actuated palm-sized crawling fall pattern of the OctoRoACH [9], the model captures the robots have been fabricated; see for example DynaRoACH [8] average behavior of the robot when crawling at low speeds and OctoRoACH [9] (Fig. 1). in a quasi-static fashion, and allows for a direct mapping Despite the introduction of a large number of small legged between model parameters and robot kinematics. robots, our understanding on how autonomous navigation can This paper validates the suitability of the considered be performed at these scale is still limited. Indeed, with few template for the OctoRoACH. The procedure involves the exceptions [10], [11], analysis is generally scarce. The few use of a motion capture system to record three curvature- available robot modeling approaches have been motivated parameterized motion primitives: (i) straight line, (ii) clock- by car-like robot methodologies. Yet, bio-inspired models wise turn, and (iii) counter-clockwise turn. Solving a con- may be more suitable for capturing intrinsic robot behaviors strained optimization problem yields nominal model param- associated with the legs. eter values that make the model’s behavior match experi- Most of the existing bio-inspired modeling approaches mentally observed robot data, on average. These data-based yield horizontal-plane reduced-order dynamical models. The primitives are then used in an RRT solver [24, Section 7.2.2] Lateral Leg Spring (LLS) model [12]–[14] offers justifica- for finding paths in environments populated with obstacles. tion for lateral stabilization [15], and is used for deriving The work in this paper is part of our effort to port naviga- turning strategies [16], [17] for hexapedal runners. The most tion and planning tools into the domain of miniature legged common configuration of the LLS consists of a rigid torso robots, and extends earlier work [10] by investigating the and two prismatic legs modeled as massless springs, each potential of bio-inspired models for navigation. The efficacy representing the collective effect of a support tripod. The of the SFM template in navigation at the miniature scale opens the way for linking high-level navigation objectives This work is supported in part by NSF under grants CMMI-1130372 and to control strategies implementable at the physical platform. CNS-1035577, and by ARL MAST CTA # W911NF-08-2-0004. The authors are with the Department of Mechanical Engineering, Uni- In principle, this template-based approach can be applied to versity of Delaware. Email: {kkaryd, liuyan, poulakas, btanner}@udel.edu a range of miniature crawling robots. II.THE SWITCHING FOUR-BAR MECHANISM the Gauss-Bonnet Theorem [25, Section 4-5]. In our planar The SFM template (Fig. 2(a)) is a horizontal-plane model configuration it results in consisting of a rigid torso and four rigid legs [23]. The L L X Z sj+1 X legs move according to the foot-fall pattern depicted in k(s)ds + χj = 2π , (1) Fig. 2(b). As the gait is executed, the torso and the legs form j=0 sj j=0 two alternating four-bar linkages, parameterized by the leg where, L is the total number of steps taken by the SFM touchdown and liftoff angles, and the leg angular velocity. template, sj is the arc length of step j, and k(s) is the θ curvature of the curve component produced at each step: 0 00 00 0 O2 x y − x y k(s) = . 0 2 0 2 3 O ((x ) + (y ) ) 2 4 φ2 B 4 2 φ4 The quantity χj is the instantaneous change in the direction 1 3 of motion of G when the model transitions from step j to step j + 1 (Fig. 3). G d O3 l 4 2 χj < 0 O1 φ3 y 1 3 φ1 A y χj+1 > 0

O x x

(a) (b) Fig. 3. Instantaneous change in the direction of motion when switching between steps. The sign of the angle is determined by the right-hand rule. Fig. 2. (a) The SFM template. It consists of two pairs of legs that become active in turns, forming two fourbar linkages, {O1A, AB, BO2} and {O3A, AB, BO4}. d is the distance between the two hip-point joints An appropriate number of steps allows the model to A and B, l denotes the leg length, and G is its geometric center. (b) The transcribe a closed circular curve, for which foot-fall pattern followed by the model. Z 2πR = ds . (2) A. Key Features c Combining (1) and (2) yields With respect to Fig. 2(a), the two leg pairs {AO ,BO }, 1 2 R and {AO ,BO } are denoted as right and left pair, respec- ds 3 4 R = c . (3) tively. It is assumed that no slipping occurs between the tips PL R sj+1 PL j=0 k(s)ds + j=0 χj of the legs and the ground, and that only one pair is active sj at all times—resulting to a 50% duty factor between legs. Then, the average path curvature produced by specific values 2 The state of the model is a tuple (xG, yG, θ) ∈ R × S, for the touchdown and liftoff angles is obtained from (3) as where (xG, yG) denotes the position of the geometric center kpath = 1/R. In principle, (3) allows one to translate “macro- of the model, G, with respect to some inertial coordinate scopic” requirements regarding desired path curvatures into frame {O}, and θ is the angle formed between the longitu- model parameters realizing them; see Section III-D below. dinal body-fixed axis and the y-inertial axis. The evolution C. Model Properties of the state during each step is determined by the kinematics of the respective active pair. Since each pair is kinematically Figure 4 graphically presents the set of reachable states 2 equivalent to a four-bar linkage, the motion at every step is from an initial state q0 ∈ C ⊂ R × S at time T , denoted fully determined by one degree of freedom, taken here to be R(q0,T ), for T = 3 seconds. Without loss of generality, the angle φ1 for the right pair, and the angle φ3 for the left. we pick q0 = (0, 0, 0) in units of [cm, cm, deg], set the The geometric characteristics of a model path strongly parameters d and l to 13 cm and 3 cm respectively, and ˙ depend on the values of the model parameters. Different choose the angular velocity of both leg pairs to φRL = 8.02 combinations of touchdown and liftoff angles can produce [deg/sec]. We also assume that all four legs are initiated with td td td td very different path profiles, the geometric features of which the same touchdown angle (i.e. φ1 = φ2 = φ3 = φ4 ). are characterized by tools from differential geometry [25]. The reachable set gives us insight into the controllability properties of the model. From the graph produced in Fig. 4, B. Characterizing the Geometry of a Model Path it follows that this system is accessible [24]. In fact, we Due to the fact that the model involves switching between can achieve small-time local accessibility if we restrict the two different four-bar mechanisms, the resulting geometric problem to R2, and treat the orientation θ as a parameter. paths are piecewise differentiable. At the switching point, however, we observe an instantaneous change in the direction III.ADAPTINGTHESFMTO OCTOROACH BEHAVIORS of motion. To calculate the curvature of such paths, produced Experimental data is used to identify the SFM parameters by a specific combination of parameters values, we use that enable the model to match the average robot behavior. To specify the target curvatures for the primitives we first 20 calculate the curvature of the tightest robot turn.1 We exper- −1 15 imentally measured it at 0.25 cm . The range of motion is thus prescribed between this turning curvature and zero, 10 y [cm] and is partitioned into four sectors. The curvature of the CW −1 5 and CCW primitives is set at 0.063 cm . The SL primitive has zero curvature, in theory; however, measurement noise, 0 robot design errors, and random ground interaction effects, -20 -10 0 10 20 x [cm] contributed to a nonzero experimentally measured curvature. Fig. 4. The set of reachable states produced by the SFM template, starting SL paths are therefore associated with curvatures less than o −1 at the initial state x0 = (0, 0) cm, θ0 = 0 , and for a time span of 3 sec. 0.01 cm (Table I). Not all possible combinations of touchdown and liftoff angles are shown; this leads to some small areas not being covered by the paths. TABLE I A LIBRARYOFPRIMITIVEBEHAVIORSFORTHEOCTOROACH A. Model–Robot Relation Motor Target Target Type Description Gains Orientation Curvature The OctoRoACH is designed to follow an alternating −1 (KL,KR) [deg] [cm ] tetrapod gait, as shown in Fig. 5(a). Legs {1, 2, 3, 4} form the “right” tetrapod, and legs {5, 6, 7, 8} form the “left” tetrapod. Clockwise o CW o (60, 20) θ ' −90 0.063 The ipsilateral legs of each tetrapod touch the ground at the 90 Turn same instant, and rotate in phase with the same angular ve- SL Straight-Line (40, 40) θ ' 0o ≤ 0.01 locity. The abstract eight-legged model of Fig. 5(b) illustrates Counter-Clockwise this coupling, based on which we combine ipsilateral legs of CCW (20, 60) θ ' 90o 0.063 90o Turn each tetrapod into a single “virtual” leg. This combination yields the SFM template (Fig. 5(c)), where contralateral virtual legs (e.g., {O ,O }) represent the collective effect 1 2 C. Experimental Construction of Motion Primitives of the tetrapod they replace (e.g. {1, 2, 3, 4}) [23]. The three primitives of Table I are realized by collecting open-loop planar position and orientation measurement data. O4 O2 5 The measured states comprise the planar position of the 1 5 B 1 B φ4 φ2 α geometric center of the robot (xG, yG), and its orientation 2 6 β θ; see Fig. 2(a). By convention, positive changes in the 2 6 orientation correspond to counter-clockwise angles θ. 7 3 G G 7 3 O1 O3 We collect data from a total of 250 paths for each α 4 8 β φ1 φ3 primitive. Data is captured with the use of a motion capture 4 A 8 A system at 100 Hz. All trials last 3 sec, and are conducted on a rubber floor mat surface. The robot is manually set into a (a) (b) (c) designated start area with an initial state set at (xG, yG, θ) = Fig. 5. Relating the SFM template to the OctoRoACH. (a) The foot-fall (0, 0, 0) [cm, cm, deg]. Initial pose errors are bounded by pattern of the robot, is an alternating tetrapod gait. Legs {1, 2, 3, 4} form the right tetrapod, and legs {5, 6, 7, 8} form the left tetrapod. (b) An eight- data statistics, and are shown in Table II. Note that these legged kinematic simplification of the gait mechanism used by the robot. error bounds account for measurement noise as well. The ipsilateral legs of each tetrapod are coupled, forming the angles α and β. This is shown here for the right tetrapod; the left tetrapod operates similarly. TABLE II (c) The SFM is recovered by grouping coupled legs within a tetrapod into a INITIAL POSE ERROR STATISTICS single virtual leg inducing the same displacement. Legs {1, 2, 3, 4} reduce to the pair {O1,O2}, while legs {5, 6, 7, 8} reduce to the pair {O3,O4}. Mean Standard Deviation Type [cm cm deg] [cm cm deg] CW (−0.156, −0.041, 1.23][0.177, 0.141, 1.37) B. A Library of Primitive Behaviors SL (−0.007, 0.027, 0.06][0.234, 0.054, 1.81) The robot has two motors, each driving the legs on its CCW (−0.322, −0.012, 2.50][0.156, 0.130, 1.23) own side. The two motor gains, KL, and KR, control the leg velocities of the left and right side, respectively. Figure 6 provides insight on the choice of the sec Then, we define three motion primitives: (i) straight line 3 duration for our data collections and the generated motion (SL), (ii) 90o clockwise turn (CW), and (iii) 90o counter- primitives. The histogram shows how individual experimen- clockwise turn (CCW); . Table I gives the gains realizing these tal paths disperse as time elapses. It can be seen that after the primitives. Note that one may choose to work with more end of the sec period, there is a very high dispersion around primitives—see [23]. The particular primitives considered 3 here capture the gross behavior of the robot while keeping 1The tightest CW and CCW turns are achieved by setting the motor gains the computational complexity of the navigation problem low. to (80, 0), and (0, 80) respectively. set of 250 paths is marked by black thick curves. Note that the platform tends to veer to the right (see Fig. 7(b)). D. Parameter Identification

Let WSL, WCW, and WCCW denote the collections of all experimental planar trajectories of the robot for the SL, CW, and CCW motion primitives, respectively, and let w denote an element in these sets. The averages of all elements for ave ave ave each set are marked with wSL , wCW , and wCCW. The model parameters to be identified are included in ¯ td ¯ td ¯ td ¯ td ¯ lo ¯ lo ¯ lo ¯ lo ¯˙ ζ = [φ1 , φ2 , φ3 , φ4 , φ1 , φ2 , φ3 , φ4 , φRL] . Superscript (n) stands for “nominal,” (td) indicates the model’s touchdown angles, while (lo) denotes liftoff angles, ˙ and φ¯RL marks the leg angular velocity, assumed to be the same for both pairs. To achieve CW turns we allow only the Fig. 6. Path dispersion as time elapses. All primitives start at the origin, left pair to be active, while right pair activation only leads to and are largely dispersed after the 3 sec trial. The z axis counts how many CCW turns. For SL paths, both pairs are active. As before, we paths are inside a particular grid square. Due to the selected grid size, some set d to 13 cm—equal to the length of the actual platform— paths may appear more than once inside a square. CW paths curve to the left of the page, while CCW paths curve to the right. and l to 3 cm. In order to identify the SFM parameters that result in tra- jectories that remain close to the experimental averages, we the experimental averages shown with black thick curves in formulate a constrained least-squares optimization problem Fig. 7. Thus, constructing motion primitives that last longer ave 2 min kpi(ζ) − wi k , i ∈ {SL, CW, CCW} , (4) is not meaningful; the variance in the experimental data ζ∈Z becomes unacceptably high. Capturing the variability due to where || · || denotes the L2 norm, and pi(ζ) indicates the the stochasticity inherent to leg-ground interaction is beyond trajectories generated by the model parameterized according the scope of this paper. Preliminary considerations appear to the value of the vector ζ. Then, the nominal parameter in [26], while a detailed account on a method for extending vector ζ for each primitive is selected as the solution of (4). deterministic models to a stochastic regime for capturing the The solution, ζˆ, enables the SFM to produce the trajectories uncertainty within the model is reported in [27]. shown in Fig. 7 with blue thick dashed curves. Table V Table III contains the average final state of the robot for contains the numerical values of the components of ζˆ, each primitive, while Table IV presents the target and average allowing SFM to capture on average the behavior of the observed values for path curvatures and final orientations. system, as shown in Fig. 7. Due to their close matching with Both averages are very close to their target values, although the experimental averages, the nominal SFM trajectories are individual paths may deviate significantly. used to calculate the values of the observed curvatures shown in Table IV by applying (3). TABLE III AVIGATION WITH THE EMPLATE FINALPOSE (AVERAGE VALUES) IV. N T Having recorded the motion primitives, we combine them x y θ Type G G [cm] [cm] [deg] to navigate in spaces populated with obstacles. To solve this problem, we employ a rapidly exploring random tree [24, CW 12.928 11.307 −85.02 Section 7.2.2] (RRT) solver that uses the constructed primi- SL 3.246 23.044 −10.62 tives in order to generate new vertices. The choice of RRT is CCW −14.183 11.960 90.90 primarily due to its popularity, proven experimental success, and the availability of off-the-shelf software implementing the basic algorithms. TABLE IV The implementation considered here requires no tuning, CURVATURE AND FINAL ORIENTATION (AVERAGE VALUES) but it does not allow for an early termination of a primitive. This means that a generated SL, CW, or CCW path is imme- Target Observed Target Observed Type Curvature Curvature Orientation Orientation diately discarded if it intersects with the obstacle region, and [cm−1] [cm−1] [deg] [deg] that only the end of each primitive can be used to define a new vertex. Because of this restriction, the algorithm may not CW 0.063 0.066 ' −90o −85.02 always find a solution. This can be rectified by introducing o SL ≤ 0.01 0.009 ' 0 −10.62 a tuning parameter that shortens execution times. As the CCW 0.063 0.068 ' 90o 90.90 number of vertices increases, a path from the initial to the goal state is found more easily. The trade-off, however, is an Finally, Fig. 7 shows 50 randomly selected paths (in increase in the computation time for the algorithm to find a magenta) for each primitive. The average of the whole data solution. 18 30 18

16 16 25 14 14

12 20 12

10 10 15 y [cm] y [cm] 8 y [cm] 8

6 10 6 4 4 5 2 2

0 0 0 -20 -15 -10 -5 0 -10 -5 0 5 10 0 5 10 15 20 x [cm] x [cm] x [cm] (a) (b) (c) Fig. 7. Experimental data for the primitives contained in Table I and the respective model output counterparts. The experimental average out of a total of 250 paths for each case is shown in black thick curves, and 50 randomly selected paths of the robot’s geometric center are shown in magenta. The blue thick dashed curves depict the output of the model parameterized by the nominal parameter values of Table V. TABLE V IDENTIFIED MODEL PARAMETERS

φtd,n φtd,n φtd,n φtd,n φlo,n φlo,n φlo,n φlo,n φ˙ Type 1 2 3 4 1 2 3 4 RL [deg] [deg] [deg] [deg] [deg] [deg] [deg] [deg] [deg/sec] CW 32.72 4.19 0 0 −55.28 −83.81 0 0 7.56 SL 56.72 45.84 68.18 70.47 −51.78 −62.66 −57.70 −55.41 9.00 CCW 0 0 26.28 1.37 0 0 −55.66 −80.57 8.65

V. RESULTS the full primitive length, even if this may not be necessary. Allowing the primitives’ execution time to vary can improve We implement the solver in four illustrative simulation the quality of paths, but it will increase computational scenarios. The initial state varies for each case (see Table VI) complexity and time required to find a solution. This trade- but the environment (depicted in Fig. 8) stays the same. off is currently being investigated. Similarly, the goal is set at q = (x , y , θ ) = (220, 220, 0) d d d d Our current work also addresses the experimental imple- [cm, cm, deg] and remains unchanged. Due to the constraints mentation and validation of the computed paths shown in of the problem and the discretization induced by the solver, Fig. 8 on the robot. This is, however, a challenging task reaching exactly q is unlikely. Thus, we accept as successful d since the high variability observed in the behavior of the all paths that end within a radius of 10 cm around (x , y ), d d robot renders successful completion of the desired paths with a final orientation in the range between −30◦ and 30◦. unlikely; our previous work in [10] illustrates this point for TABLE VI a morphologically similar robot. For the robot considered INITIAL STATES FOR CASE STUDY SIMULATIONS here, the current low-level hardware does not allow for integration of on-board sensors (e.g. a compass and IMU) Case (a) (b) (c) (d) for state feedback, hindering the on-board implementation xG [cm] 40 120 120 120 of feedback control policies.

yG [cm] 10 10 10 10 VI.CONCLUSIONS θ [deg] 0 0 −45 45 We demonstrate that the SFM template can be effectively used for navigation of miniature legged robots such as the With respect to Fig. 8, the actual obstacles are marked OctoRoACH, when the platform operates in a quasi-static in blue, while light gray is used to denote the portion of fashion at low crawling speeds. Combining this kinematic the configuration space where the boundary of the model template with a generic RRT solver yields reasonably good touches, or crosses the boundary of the obstacle (“grown performance in terms of finding a path from an initial to a obstacle”). We use magenta to color the branches of the final state within cluttered environments. The paths produced constructed trees, and we finally highlight in red the sequence respect the kinematic constraints of the robot, with minimal of primitives that leads from the initial to a neighborhood of modifications and tuning. the desired state, marked with a large black circle. Motion capabilities and mobility constraints are captured Figure 8 suggests that the solver finds a solution for all in the form of motion primitives, and encode basic motion cases presented. Note, however, that paths may become very behaviors: straight line, 90o clockwise and counter-clockwise curvy toward the end due to the desired final orientation we turns. The parameters of these primitives are given by the have chosen. This behavior is exacerbated by the execution solution of a constrained optimization problem, formulated time of primitives being fixed. The solver always chooses on a set of experimentally collected data from observed robot 240 240 240 240

200 200 200 200

160 160 160 160

120 120 120 120 y [cm] y [cm] y [cm] y [cm]

80 80 80 80

40 40 40 40

0 0 0 0 0 40 80 120 160 200 240 0 40 80 120 160 200 240 0 40 80 120 160 200 240 0 40 80 120 160 200 240 x [cm] x [cm] x [cm] x [cm] (a) (b) (c) (d)

Fig. 8. Navigation scenarios for different initial states. The first case differs in the position, while the last three start at the same point, but with different initial orientations. Obstacles are at the same location for all cases. Irrespective of the initial state though, we were always able to find a solution. The curvy final part in cases (a), (c), and (d) is due to the desired final orientation being set in the interval [−30, 30]o; the effect can be remedied by letting the execution time vary.

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